A Review of First-Passage Theory for the Segerdahl-Tichy Risk Process and Open Problems

Florin Avram and Jose-Luis Perez-Garmendia
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Florin Avram: Laboratoire de Mathématiques Appliquées, Université de Pau, 64000 Pau, France
Jose-Luis Perez-Garmendia: Centro de Investigación en Matemáticas, Guanajuato 36020, Mexico

Risks, 2019, vol. 7, issue 4, 1-21

Abstract: The Segerdahl-Tichy Process, characterized by exponential claims and state dependent drift, has drawn a considerable amount of interest, due to its economic interest (it is the simplest risk process which takes into account the effect of interest rates). It is also the simplest non-Lévy, non-diffusion example of a spectrally negative Markov risk model. Note that for both spectrally negative Lévy and diffusion processes, first passage theories which are based on identifying two “basic” monotone harmonic functions/martingales have been developed. This means that for these processes many control problems involving dividends, capital injections, etc., may be solved explicitly once the two basic functions have been obtained. Furthermore, extensions to general spectrally negative Markov processes are possible; unfortunately, methods for computing the basic functions are still lacking outside the Lévy and diffusion classes. This divergence between theoretical and numerical is strikingly illustrated by the Segerdahl process, for which there exist today six theoretical approaches, but for which almost nothing has been computed, with the exception of the ruin probability. Below, we review four of these methods, with the purpose of drawing attention to connections between them, to underline open problems, and to stimulate further work.

Keywords: Segerdahl process; affine coefficients; first passage; spectrally negative Markov process; scale functions; hypergeometric functions (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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